Likelihood Ratios for Mixtures: Binary Approach · I thank Simone Gittelson, Bruce Weir and John...
Transcript of Likelihood Ratios for Mixtures: Binary Approach · I thank Simone Gittelson, Bruce Weir and John...
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 1
Likelihood Ratios for Mixtures: Binary Approach
AcknowledgementI thank Simone Gittelson, Bruce Weir and John Buckleton for their helpful discussions.
DisclaimerPoints of view in this presentation are mine and do not necessarily represent the official position or policies of the National Institute of Standards and Technology.
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Likelihood Ratio
7 8 9 10 {8,9}
person of interest (POI)
Is the POI a contributor to this
DNA mixture?
Likelihood Ratioperson of interest (POI)
The POI is a contributor.
The POI is not a contributor.
7 8 9 10
1) What is the probability of obtaining these DNA typingresults if the POI is a contributor?
2) What is the probability of obtaining these DNA typingresults if the POI is not a contributor?
{8,9}
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Likelihood Ratioperson of interest (POI)
7 8 9 10
1) What is the probability of obtaining these DNA typingresults if the POI is a contributor?
contributors: POI and someone else
1 × 2𝑝7𝑝10
{8,9}
Likelihood Ratio
7 8 9 10
2) What is the probability of obtaining these DNA typing resultsif the POI is not a contributor?
Contributor 1 Contributor 2
7,8 9,10 2𝑝7𝑝8 × 2𝑝9𝑝10
7,9 8,10 2𝑝7𝑝9 × 2𝑝8𝑝10
7,10 8,9 2𝑝7𝑝10 × 2𝑝8𝑝9
8,9 7,10 2𝑝8𝑝9 × 2𝑝7𝑝10
8,10 7,9 2𝑝8𝑝10 × 2𝑝7𝑝9
9,10 7,8 2𝑝9𝑝10 × 2𝑝7𝑝8
= 24𝑝7𝑝8𝑝9𝑝10
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Likelihood Ratio
{8,9}
person of interest (POI)
7 8 9 10 24𝑝7𝑝8𝑝9𝑝10
2𝑝7𝑝10
=1
12𝑝8𝑝9= 8.3333if 𝑝8 = 0.1 and 𝑝9 = 0.1
It is 8.3333 times more probable to obtain these DNA typing resultsif the POI is a contributor than if the POI is not a contributor.
EPG of the crime stain:
BU Mixture (http://www.bu.edu/dnamixtures/) ID_2_SCD_NG0.5_R4,1_A1_V1
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Standpoints of the prosecution and the defense
𝐻𝑝: The DNA came from the POI and an unknown contributor.
𝐻𝑑: The DNA came from two unknown contributors.
EPG of the crime stain
person of interest
(POI)
Person of interest (POI)
D8 13,16
D21 28,28
D7 8,12
CSF1PO 12,12
D3 16,16
TH01 7,9.3
D13 12,13
D16 12,13
D2 23,25
D19 13,13
vWA 15,19
TPOX 11,11
D18 14,20
D5 11,13
FGA 20,28
EPG of the crime stain:
Boston University Mixture (http://www.bu.edu/dnamixtures/): ID_2_SCD_NG0.5_R4,1_A1_V1
𝐺𝐶𝑆 𝐺𝑃𝑂𝐼
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Likelihood Ratio (LR)
𝐿𝑅 =Pr(𝐺𝐶𝑆|𝐺𝑃𝑂𝐼 , 𝐻𝑝)
Pr(𝐺𝐶𝑆|𝐺𝑃𝑂𝐼 , 𝐻𝑑)divided by
the probability of observing the DNA typing results of the criem staingiven the POI’s genotype and that the DNA came from two unknowncontributors.
The probability of observing the DNA typing results of the crime staingiven the POI’s genotype and that the DNA came from the POI and one unknown contributor
numerator
denominator
Note that the conditioning information 𝐼 has been omitted in the above equation to focus your attention on 𝐺𝐶𝑆 and 𝐺𝑃𝑂𝐼 .
Person of interest (POI)
D8 13,16
D21 28,28
D7 8,12
CSF1PO 12,12
D3 16,16
TH01 7,9.3
D13 12,13
D16 12,13
D2 23,25
D19 13,13
vWA 15,19
TPOX 11,11
D18 14,20
D5 11,13
FGA 20,28
EPG of the crime stain:
Boston University Mixture (http://www.bu.edu/dnamixtures/): ID_2_SCD_NG0.5_R4,1_A1_V1
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𝑝14 = 0.134𝑝16 = 0.147𝑝18 = 0.078𝑝20 = 0.018
Likelihood Ratio (LR)
D18S51
𝐺𝑃𝑂𝐼 = {14,20}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is a contributor and the POI has genotype{14,20}?
Numerator:
Major Minor
16,18 14,20
𝑃𝑟 16,18 × 𝑃𝑟 14,20
= 2𝑝16𝑝18 × 1
= 2𝑝16𝑝18
Likelihood Ratio (LR)
D18S51
𝐺𝑃𝑂𝐼 = {14,20}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is not a contributor?
Denominator:
Major Minor
16,18 14,20
𝑃𝑟 16,18 × 𝑃𝑟 14,20
= 2𝑝16𝑝18 × 2𝑝14𝑝20
= 4𝑝14𝑝16𝑝18𝑝20
𝑝14 = 0.134𝑝16 = 0.147𝑝18 = 0.078𝑝20 = 0.018
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Likelihood Ratio (LR)
D18S51𝑝14 = 0.134𝑝16 = 0.147𝑝18 = 0.078𝑝20 = 0.018
𝐺𝑃𝑂𝐼 = {14,20}
𝐿𝑅 =2𝑝16𝑝18
4𝑝14𝑝16𝑝18𝑝20
=1
2𝑝14𝑝20
= 207.30
Likelihood Ratio (LR)
D18S51
𝐺𝑃𝑂𝐼 = {14,20}
The DNA typing results are 207 times more probable if the DNA came from the person of interest and an unknown contributor than if
the DNA came from two unknown contributors.
𝑝14 = 0.134𝑝16 = 0.147𝑝18 = 0.078𝑝20 = 0.018
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Unrestricted LR - Peak Heights are ignored
𝑝14 = 0.134𝑝16 = 0.147𝑝18 = 0.078𝑝20 = 0.018
Likelihood Ratio (LR)
D18S51
𝐺𝑃𝑂𝐼 = {14,20}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is a contributor and the POI has genotype{14,20}?
Numerator:
𝑃𝑟 16,18 × 𝑃𝑟 14,20
= 2𝑝16𝑝18 × 1
= 2𝑝16𝑝18
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Likelihood Ratio (LR)
D18S51
𝐺𝑃𝑂𝐼 = {14,20}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is not a contributor?
Denominator:
𝑝14 = 0.134𝑝16 = 0.147𝑝18 = 0.078𝑝20 = 0.018
Think of it like this!
Likelihood Ratio
14 16 18 20
2) What is the probability of obtaining these DNA typing resultsif the POI is not a contributor?
Contributor 1 Contributor 2
14,16 18,20 2𝑝14𝑝16 × 2𝑝18𝑝20
14,18 16,20 2𝑝14𝑝18 × 2𝑝16𝑝20
14,20 16,18 2𝑝14𝑝20 × 2𝑝16𝑝18
18,20 14,16 2𝑝18𝑝20 × 2𝑝14𝑝20
16,20 14,18 2𝑝16𝑝20 × 2𝑝14𝑝18
16,18 14,20 2𝑝16𝑝18 × 2𝑝14𝑝20
= 24𝑝14𝑝16𝑝18𝑝20
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24𝑝14𝑝16𝑝18𝑝20
2𝑝16𝑝18
12𝑝14𝑝20
1
=
𝐿𝑅 =2𝑝16𝑝18
4𝑝14𝑝16𝑝18𝑝20
=1
2𝑝14𝑝20
= 207.30
Restricted LR Unrestricted LR
= 34.55
= 0.0289
1
1
= 34.55
12𝑝14𝑝20
Person of interest (POI)
D8 13,16
D21 28,28
D7 8,12
CSF1PO 12,12
D3 16,16
TH01 7,9.3
D13 12,13
D16 12,13
D2 23,25
D19 13,13
vWA 15,19
TPOX 11,11
D18 14,20
D5 11,13
FGA 20,28
EPG of the crime stain:
Boston University Mixture (http://www.bu.edu/dnamixtures/): ID_2_SCD_NG0.5_R4,1_A1_V1
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Likelihood Ratio (LR)
CSF1PO
𝐺𝑃𝑂𝐼 = {12,12}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is a contributor and the POI has genotype{12,12}?
Numerator:
Major Minor
10,11 12,12
10,11 10,12
10,11 11,12
𝑃𝑟 10,11 × 𝑃𝑟 12,12
= 2𝑝10𝑝11 × 1
= 2𝑝10𝑝11
𝑝10 = 0.220𝑝11 = 0.309𝑝12 = 0.360
allele at 12 is above the stochastic threshold
Likelihood Ratio (LR)
CSF1PO
What is the probability of obtaining these DNA typing results for the crime stain if the POI is not a contributor?
Denominator:
Major Minor
10,11 12,12
10,11 10,12
10,11 11,12
𝑝10 = 0.220𝑝11 = 0.309𝑝12 = 0.360
𝐺𝑃𝑂𝐼 = {12,12}
allele at 12 is above the stochastic threshold
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Likelihood Ratio (LR)
CSF1PO
Denominator:
Major Minor
10,11 12,12
10,11 10,12
10,11 11,12
𝑃𝑟 10,11 × 𝑃𝑟 12,12 + 𝑃𝑟 10,11 × 𝑃𝑟 10,12 + 𝑃𝑟 10,11 × 𝑃𝑟 11,12
= 2𝑝10𝑝11 × 𝑝122 + 2𝑝10𝑝11 × 2𝑝10𝑝12
+2𝑝10𝑝11 × 2𝑝11𝑝12
= 2𝑝10𝑝11𝑝12(𝑝12 + 2𝑝10 + 2𝑝11)
𝑝10 = 0.220𝑝11 = 0.309𝑝12 = 0.360
𝐺𝑃𝑂𝐼 = {12,12}
allele at 12 is above the stochastic threshold
Likelihood Ratio (LR)
CSF1PO
𝐿𝑅 =2𝑝10𝑝11
2𝑝10𝑝11𝑝12(𝑝12 + 2𝑝10 + 2𝑝11)
=1
𝑝12 𝑝12 + 2𝑝10 + 2𝑝11
= 1.96
𝑝10 = 0.220𝑝11 = 0.309𝑝12 = 0.360
𝐺𝑃𝑂𝐼 = {12,12}
allele at 12 is above the stochastic threshold
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Likelihood Ratio (LR)
CSF1PO
The DNA typing results are about 2 times more probable if the DNA came from the
person of interest and an unknown contributor than if the DNA came from two
unknown contributors.
𝑝10 = 0.220𝑝11 = 0.309𝑝12 = 0.360
𝐺𝑃𝑂𝐼 = {12,12}
allele at 12 is above the stochastic threshold
Person of interest (POI)
D8 13,16
D21 28,28
D7 8,12
CSF1PO 12,12
D3 16,16
TH01 7,9.3
D13 12,13
D16 12,13
D2 23,25
D19 13,13
vWA 15,19
TPOX 11,11
D18 14,20
D5 11,13
FGA 20,28
EPG of the crime stain:
Boston University Mixture (http://www.bu.edu/dnamixtures/): ID_2_SCD_NG0.5_R4,1_A1_V1
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Allele 28 is below the stochastic threshold
50 RFUs
150 RFUs
Analytical Threshold (AT)
Stochastic Threshold (ST)
Noise
Peak real, but not used for CPI
Peak real, can be used for CPI –Confidence of being Homozygous
Peak not considered
reliable
Example values (empirically determined based on own internal validation)
(Reporting/NoiseLimit-of-Detection)
(Match Interpretation/ Dropout/Reporting)
Different Thresholds Used with CE Data
“2p” Rule
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Likelihood Ratio (LR)
D21S11
𝐺𝑃𝑂𝐼 = {28,28}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is a contributor and the POI has genotype{28,28}?
Numerator:
Major Minor
30,32.2 28,F
𝑃𝑟 30,32.2 × 𝑃𝑟 28, 𝐹
= 2𝑝30𝑝32.2 × 1
= 2𝑝30𝑝32.2
𝑝28 = 0.159𝑝30 = 0.283𝑝32.2= 0.090
Allele 28 is below the stochastic threshold
Likelihood Ratio (LR)
D21S11
𝐺𝑃𝑂𝐼 = {28,28}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is not a contributor?
Denominator:
𝑃𝑟 30,32.2 × 𝑃𝑟 28, 𝐹
= 2𝑝30𝑝32.2 × [2𝑝28 1 − 𝑝28 + 𝑝282 ]
= 2𝑝30𝑝32.2(2𝑝28 − 𝑝282 )
𝑝28 = 0.159𝑝30 = 0.283𝑝32.2= 0.090
Major Minor
30,32.2 28,F
Allele 28 is below the stochastic threshold
“2p” or p2 + 2p(1-p)
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Derivation of the 2p Rule
• Two ways to think of it…
2pq 2pq 2pq
(easy)
“2p” or p2 + 2p(1-p)
Derivation of the 2p Rule
• Two ways to think of it…
5 allele system - P Q R S T
(mathematical)
“P” is below ST
P
ANY genotype with a “P” allele cannot be excluded
PP PQ PR PS PT
“2p” or p2 + 2p(1-p)
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5 allele system - P Q R S T
“P” is below ST
P
ANY genotype with a “P” allele cannot be excluded
PP PQ PR PS PT
Prob = p2 + 2pq + 2pr + 2ps + 2pt
Prob = p2 + 2p(q + r + s + t)
Prob = p2 + 2p(1-p)
Prob = p2 + 2p - 2p2
Prob = 2p - p2
“2p” or p2 + 2p(1-p)
Likelihood Ratio (LR)
D21S11
𝐺𝑃𝑂𝐼 = {28,28}
𝐿𝑅 =2𝑝30𝑝32.2
2𝑝30𝑝32.2(2𝑝28 − 𝑝282 )
=1
2𝑝28 − 𝑝282
= 3.42
𝑝28 = 0.159𝑝30 = 0.283𝑝32.2= 0.090
Allele 28 is below the stochastic threshold
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What does a 𝑳𝑹 ≈ 𝟑mean?A. The person of interest committed the
crime.
B. A total of 3 peaks were observed at this locus.
C. It is about 3 times more probable that the DNA came from the person of interest and an unknown contributor than that the DNA came from two unknown contributors.
D. There are 3 contributors to this DNA mixture.
E. The DNA typing results are about 3 times more probable if the DNA came from the person of interest and an unknown contributor than if the DNA came from two unknown contributors.
A. B. C. D. E. F.
0% 0% 0%0%0%0%
Response Counter
Person of interest (POI)
D8 13,16
D21 28,28
D7 8,12
CSF1PO 12,12
D3 16,16
TH01 7,9.3
D13 12,13
D16 12,13
D2 23,25
D19 13,13
vWA 15,19
TPOX 11,11
D18 14,20
D5 11,13
FGA 20,28
EPG of the crime stain:
Boston University Mixture (http://www.bu.edu/dnamixtures/): ID_2_SCD_NG0.5_R4,1_A1_V1
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Likelihood Ratio (LR)
TPOX
𝐺𝑃𝑂𝐼 = {11,11}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is a contributor and the POI has genotype{11,11}?
Numerator:
Major Minor
8,8 11,11
8,8 8,11
𝑃𝑟 8,8 × 𝑃𝑟 11,11
= ⋯
ST
The peak at 11 is abovethe stochastic threshold.
𝑝8= 0.525𝑝11 = 0.252
Likelihood Ratio (LR)
TPOX
𝐺𝑃𝑂𝐼 = {11,11}
What is the probability of obtaining these DNA typing results for the crime stain if the POI is not a contributor?
Denominator:
𝑃𝑟 8,8 × 𝑃𝑟 11,11 + 𝑃𝑟 8,8 × 𝑃𝑟 8,11
= ⋯
𝑝8= 0.525𝑝11 = 0.252
Major Minor
8,8 11,11
8,8 8,11
ST
The peak at 11 is abovethe stochastic threshold.
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What is the likelihood ratio?
A.𝑝82
𝑝82 𝑝112 +2𝑝8𝑝11
=1
𝑝11 𝑝11+2𝑝8
B.1
𝑝11+2𝑝8
C. 1
D.1
2𝑝8𝑝11
E.1
𝑝112
F. infinity
G. ???
A. B. C. D. E. F. G.
0% 0% 0% 0%0%0%0%Response Counter
ST
The peak at 11 is abovethe stochastic threshold.
Likelihood Ratio (LR)
TPOX
𝐺𝑃𝑂𝐼 = {11,11}
ST
𝐿𝑅 =𝑝82
𝑝82𝑝112 + 𝑝8
22𝑝8𝑝11
=1
𝑝11(𝑝11 + 2𝑝8)
= 3.05
𝑝8= 0.525𝑝11 = 0.252
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Likelihood Ratio (LR)
TPOX
𝐺𝑃𝑂𝐼 = {11,11}
ST
The DNA typing results are about 3 times more probable if the DNA came from the
person of interest and an unknown contributor than if the DNA came from two
unknown contributors.
𝑝8= 0.525𝑝11 = 0.252
Likelihood Ratio (LR) for all loci
𝐻𝑝: The DNA came from the POI and an unknown contributor.
𝐻𝑑: The DNA came from two unknown contributors.
If 𝐻𝑝 is true, is the POI the major contributor or the minor contributor?
If 𝐻𝑝 is true, the POI could be either the major contributor or
the minor contributor. Let us consider these possibilities to be
equally probable. So if 𝐻𝑝 is true, there is a probability of 1
2
that the POI is the major contributor and a probability of 1
2
that the POI is the minor contributor.
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We can only observe these DNA typing results if the POI is the minor contributor.
Major Minor
8,8 11,11
8,8 8,11
Major Minor
30,32.2 28,F
𝐺𝑃𝑂𝐼 = {11,11}
𝐺𝑃𝑂𝐼 = {28,28}
Major Minor
10,11 12,12
10,11 10,12
10,11 11,12
Major Minor
16,18 14,20
𝐺𝑃𝑂𝐼 = {12,12}
𝐺𝑃𝑂𝐼 = {14,20}
TPOX:
D21S11:
CSF1PO:
D18S51:
Likelihood Ratio (LR) for all loci
𝐻𝑝: The DNA came from the POI and an unknown contributor.
𝐻𝑑: The DNA came from two unknown contributors.
Numerator:
Because these DNA typing results are only possible when the POI is the minor contributor, and the POI is the minor
contributor with a probability of 1
2, we multiply the numerator
of the likelihood ratio for the entire profile by 1
2.
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Locus Likelihood Ratio
D8S1179 3.66
D21S11 3.42
D7S820 3.74
CSF1PO 1.96
D3S1358 2.39
TH01 1.75
D13S317 4.58
D16S539 1.89
D2S1338 5.03
D19S433 1.29
vWA 1
TPOX 3.05
D18S51 207.30
D5S818 3.77
FGA 1
Likelihood Ratio (LR)
All Loci: 𝑳𝑹 = 𝟐. 𝟓 × 𝟏𝟎𝟕
True or false?
A likelihood ratio of 𝟐. 𝟓 × 𝟏𝟎𝟕 means that it is 𝟐. 𝟓 ×𝟏𝟎𝟕 times more probable that the DNA came from the person of interest and an unknown contributor than that the DNA came from two unknown contributors.
A. True
B. False
True
False
0%0%Response Counter
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Likelihood Ratio (LR)
𝑳𝑹 = 𝟐. 𝟓 × 𝟏𝟎𝟕 = 𝟐𝟓𝒎𝒊𝒍𝒍𝒊𝒐𝒏
The DNA typing results are about 25 million times more probable if the DNA
came from the person of interest and an unknown contributor than if the DNA
came from two unknown contributors.
Factor of 2
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Suppose…
What is the probability you will randomly pick a red marble?
Pr(picking left) x Pr(left is red) + Pr(picking right) x Pr(right is red)
0.5 x 1 + 0.5 x 1= 1
Suppose…
What is the probability you will randomly pick a red marble?
Pr(picking left) x Pr(left is red) + Pr(picking right) x Pr(right is red)
0.5 x 1 + 0.5 x 0= 0.5
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Now Suppose…
A B C D {A, B}
person of interest (POI)
Hp: POI and someone else
Hd: DNA came from 2 unknowns
A B C D
{A, B}
person of interest (POI)
Only 2 genotype combinations are possible for Hp
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 28
Row 1 - 0.5 x 1 x 2pc pd
Row 6 - 0.5 x 2pc pd x 1
= 0.5 x 1 x 2pc pd + 0.5 x 2pc pd x 1
= 2pc pd
Hp
7 8 9 10
2) What is the probability of obtaining these DNA typing resultsif the POI is not a contributor?
Contributor 1 Contributor 2
7,8 9,10 2𝑝7𝑝8 × 2𝑝9𝑝10
7,9 8,10 2𝑝7𝑝9 × 2𝑝8𝑝10
7,10 8,9 2𝑝7𝑝10 × 2𝑝8𝑝9
8,9 7,10 2𝑝8𝑝9 × 2𝑝7𝑝10
8,10 7,9 2𝑝8𝑝10 × 2𝑝7𝑝9
9,10 7,8 2𝑝9𝑝10 × 2𝑝7𝑝8
= 24𝑝7𝑝8𝑝9𝑝10
For the Hd (recall this?)
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 29
Likelihood Ratio
2pc pd
24pa pb pc pd
1
12pa pb
=
This is like the box with 2 red marbles: 0.5 + 0.5 = 1 in the numerator
Anytime you have a “mirror image” for the numerator, there is nofactor of 0.5 (or factor of 2 in the denominator)
Now Suppose…
A B C D {A, B}
person of interest (POI)
Hp: POI and someone else
Hd: DNA came from 2 unknowns
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 30
A B C D
{A, B}
person of interest (POI)
Hp: POI and someone else
Hd: DNA came from 2 unknowns
Hp = 0.5 x 1 x 2pc pd = 0.5 x 2pc pd
Hd = 2pa pb x 2pc pd
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 31
Likelihood Ratio
0.5 x 2pc pd
4pa pb pc pd
0.5
2pa pb
=
This is like the box with 1 red, 1 blue marble: 0.5 (left)+ 0.5 (right)
Therefore, a factor of 0.5 appears in the numerator(or factor of 2 in the denominator)
1
4pa pb
=
So - why do we even need probabilistic genotyping?
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 32
Suspect
Evidence
Suspect
Evidence
LR1
2pq=
Suspect
Evidence
“2p”
LR0
2pq= LR
?
2pq=
The Binary LR approach
The “2p” Rule
Stain = AA
Suspect = AA
ST
LR = 5LR = 100f(a) = 0.10 1/p2 = 100 1/2p = 5
NEAFS Probabilistic DNA Mixture Interpretation Workshop
9/25/2015
Cedar Crest CollegeForensic Science Training Institute 33
The “2p” Rule
Stain = AA
Suspect = AB
ST
LR = 5Exclusionf(a) = 0.10 1/2p = 5
Whatever way uncertainty is approached, probability is the only sound way to think about it.
-Dennis Lindley